Representations of superconformal algebras and mock theta functions

0.62 MB·

by  Victor G. Kac

Representations of superconformal algebras and mock theta
7 functions
1
0
2 Victor G. Kac∗ and Minoru Wakimoto†

n
a
J

2
1
To Ernest Borisovich Vinberg on his 80th birthday

]
T
R Abstract
.
h
Itiswellknownthatthenormaizedcharactersofintegrablehighestweightmodulesofgivenlevel
t
a over an affine Lie algebra g span an SL (Z)-invariant space. This result extends to admissible
m 2
g-modules, where g is a simple Lie algebra or osp . Applying the quantum Hamiltonian
1|n
[
reduction (QHR) to admisbsible g-modules when g = sl2 (resp. = osp1|2) one obtains minimal
1 bseries modules over the Virasoro (resp. N = 1 superconformal algebras), which form modular
v
4 invariant families. b
4 Another instance of modular invariance occurs for boundary level admissible modules, in-
3 cluding when g is a basic Lie superalgebra. For example, if g = sl (resp. = osp ), we
3 2|1 3|2
thus obtain modular invariant families of g-modules, whose QHR produces the minimal series
0
. modules for the N = 2 superconformal algebras (resp. a modular invariant family of N = 3
1
0 superconformal algebra modules). b
7 However, inthecasewhengisabasicLiesuperalgebradifferentfromasimpleLiealgebraor
1
osp , modular invariance of normalized supercharacters of admissibleg-modules holds outside
: 1|n
v
of boundary levels only after their modification in the spirit of Zwegers’ modification of mock
i
X theta functions. Applyingthe QHR,we obtain families of representationbs of N = 2,3,4 and big
r N = 4 superconformal algebras, whose modified (super)characters span an SL (Z)-invariant
a 2
space.
0 Introduction
Letgbeabasicfinite-dimensionalLiesuperalgebra,i.e. gissimple, its even partg isreductive
¯0
and g carries a non-degenerate invariant supersymmetric bilinear form (.|.). Given a nilpotent
element f ∈ g and K ∈ C, one associates to the triple (g,f,K) a vertex algebra WK(g,f),
¯0
obtainedfromtheuniversalaffinevertex algebraVK(g)oflevel K bythequantumHamiltonian
∗Department of Mathematics, M.I.T, Cambridge, MA 02139, USA. Email: [email protected] Supported in
part byan NSFgrant.
†Email: [email protected] . Supportedin part byDepartment of Mathematics, M.I.T.
1
reduction(QHR)[FF90],[KRW03],[KW04]. ThevertexalgebraWK(g,f)isfreelyandstrongly
generated by quantum fields, labeled by a basis of the centralizer gf of f in g.
Moreover, QHR provides a functor from the category of VK(g)-modules to the category of
WK(g,f)-modules (some modules may go to zero). This functor allows, in particular, derive
the characters of WK(g,f)-modules and their modular transformations from that of VK(g)-
modules [FKW92], [KRW03], [A05].
ThesimplestsubclassoftheW-algebrasWK(g,f), studiedindetailin[KW04],corresponds
to f = e , the lowest root vector of g for some choice of positive roots. The vertex algebra
−θ
WK(g,e ) carries a conformal vector L with central charge
−θ
Ksdim g
(0.1) c(K) = −6K +h∨−4,
K +h∨
provided that K 6= −h∨. Throughout the paper h∨, called the dual Coxeter number of g, is
the half of the eigenvalue in the adjoint representation of the Casimir element of g, for the
normalization of the bilinear form by
(0.2) (θ|θ)= 2.
This class of W-algebras, called minimal, covers all well-known superconformal algebras:
WK(sℓ ,e ) is the Virasoro vertex algebra,
2 −θ
WK(spo ,e ) is the Neveu-Schwarz algebra,
2|3 −θ
WK(sℓ ,e ) is the N = 2 superconformal algebra,
2|1 −θ
WK(psℓ ,e ) is the N =4 superconformal algebra,
2|2 −θ
WK(spo ,e ) (tensored with one fermion) is the N = 3 superconformal algebra,
2|3 −θ
WK(D(2,1;a),e ) (tensored with four fermions and one boson) is the big N = 4 super-
−θ
conformal algebra.
ThroughoutthepaperbyN superconformalalgebrawealwaysmeanthecorrespondingminimal
W-algebra.
Recall that an irreducible highest weight module L(Λ) over the affine Lie superalgebra
g is called integrable if all root vectors of g, attached to roots α, such that (α|α) > 0, act
locally nilpotently on L(Λ) (throughout the paper we keep the usual normalization (0.2) of the
binvariant bilinear form). It was established inb[KP84] that if g is a Lie algebra, then the span of
the normalized characters of integrable g-modules L(Λ) of given level K is SL(2,Z)-invariant,
since normalized numerators can be expressed in terms of theta functions (= Jacobi forms),
due to the Weyl-Kac character formulab [K90]. (The SL(2,Z)-invariance of the normalized
denominator easily follows from the Jacobi triple product identity.)
As we have discovered in [KW88], [KW89], modular invariance of normalized characters for
affine Lie algebra g holds for a much larger class of modules L(Λ), which we called admissible
modules (and we conjectured that these are all L(Λ) with the modular invariance property,
which we were ablbe to verify only for g = sℓ2). Roughly speaking, a g-module L(Λ) is called
admissible, if the Q-span of coroots of g coincides with that of the Λ-integral coroots, and with
b
b 2
respect to the corresponding affine Lie algebra g the weight Λ becomes integrable after a shift
Λ
by the Weyl vectors. We showed in [KW88] that a formula, similar to the Weyl-Kac character
formula holds if Λ is an admissible weight: onebjust has to replace in the numerator the Weyl
group of g by the subgroup, generated by reflections with respect to non-isotropic Λ-integral
coroots. It follows that the numerators of normalized admissible characters are again expressed
as linear cbombinations of Jacobi forms, which again implies modular invariance of normalized
characters of admissible g-modules.
Itwasshownin[FF90]and[FKW92]thattheQHRofadmissiblerepresentationsoftheaffine
Lie algebra g, associatedbto a simple Lie algebra g, yields all minimal series representations of
the corresponding W-algebra WK(g,f), when f is a principal nilpotent element. In particular,
for g = sℓ2b, f = pincipal (= minimal) nilpotent element, one obtains the minimal series
representations of the Virasoro algebra, and at the same time shows that their normalized
characters are modular invariant. (Note that all integrable g-modules go to zero under the
QHR if g is a Lie algebra.)
If g is a Lie superalgebra, which is not a Lie algebra, tbhe situation is different in sev-
eral respects. First, though the normalized affine superdenominator is modular invariant, the
normalized denominator is not. However, on the span of the normalized denominator and
superdenominator and their Ramond twisted analogues the group SL(2,Z) acts by monomial
matrices. Hencethequestionofmodularinvariancereducestothatofthespanofthenormalized
numerators and supernumerators and their Ramond twisted analogues.
Recall that the defect of a basic Lie superalgebra g is the cardinality of a maximal isotropic
set of roots T, i.e. a set of linearly independent pairwise orthogonal isotropic roots of g (= di-
mensionofamaximalisotropicsubspaceintheQ-spanofroots)[KW94]. Ashasbeenexplained
in[KW16], onecanexpectmodularinvariance ofnumerators ofnormalized (super)characters of
integrable (and the corresponding admissible) g-modules, even after their modification, only if
(Λ+ρ|T) = 0 for a set T, consisting of defect(g) linearly independentisotropic pairwise orthog-
onal roots. Such L(Λ) are called maximally atbypical. Provided that T is contained in a set of
simplbe roots of g, such L(Λ) are called tame. The normalized numerators of (super)characters
of admissible g- modules , corresponding to tame integrable modules, and hence their Ramond
twisted analogubes, can be expressed in terms of mock theta functions. This is a conjecture,
which has beebn verified in many cases [KW01], [KW16], [GK15]. Moreover these mock theta
functions can be modified, so that the resulting normalized (super)characters span a modular
invariant space [KW16].
In the case of g = spo , which is the only basic non Lie algebra of defect 0, the normalized
n|1
(super)numerators and their Ramond twisted analogues are again linear combinations of theta
functions, and their span is modular invariant for all the admissible respresentations [KW88].
In particular, this holds for the Lie superalgebra g = spo , when the QHR of admissible g-
2|1
modules and their twisted analogues produces all minimal series representations of the Neveu-
Schwarz andRamond algebra [KRW03], whosenormalized characters and supercharacters spabn
a modular invariant space.
The level K of an admissible g-module is a rational number, expressed via the level m of
the corresponding integrable module by
b
m+h∨
(0.3) K = −h∨, where M ∈ Z .
≥1
M
3
Of special interest are the so called boundary level g-modules, i.e. the admissible g-modules,
corresponding to the trivial (integrable) module; their levels are
b b
h∨
(0.4) K = −h∨, where M ∈ Z .
≥1
M
In this case the (super)characters and their twisted analogues can be expressed via the affine
(super)denominator, see [KW89] (resp. [GK15]) in the Lie algebra (resp. superalgebra) case.
As a result, modular invariance still holds, and therefore it holds after the QHR.
Inparticular, inthecaseg = sℓ ,whenthecorrespondingminimalW-algebraistheN = 2
2|1
superconformal algebra, the boundary levels are K = 1 −1, where M ∈ Z . Then, by the
M ≥2
QHR, one gets modular invariant representations of the N = 2 superconformal algebras with
central charge c = 3 − 6 [RY87]. (For M = 1 the g-module is trivial, hence goes to zero.)
M
These are the well-known N = 2 minimal series representations, whose modular invariance
is well known [KW94], [KRW03]. However, there abre many more tame integrable and the
corresponding admissible g-modules; their levels are [KW14], [KW15]:
m+1
(0.5) K = b−1, where M ∈ Z , m ∈ Z , gcd(2m+2,M) = 1.
≥2 >0
M
As shown in [KW14], [KW15], though the normalized (super)characters of the tame integrable
g-modules and their Ramond twisted analogues are not modular invariant, their modifications
in the spirit of Zwegers [Z08] are. Consequently, the modified (super)characters of the corre-
bsponding, via the QHR, representations of the N = 2 superconformal algebras with central
charge c(K) = 3(1− 2m+2) span a modular invariant subspace.
M
Likewise, in the case g = spo , for the boundary level
2|3
1 1
K = − , where M ∈ Z , odd,
≥1
2M 2
the QHR produces modular invariant representations of the N = 3 superconformal algebras
with central charge c = − 3 , see [KW15], Section 6. However, there are many more admissible
M
g-modules, corresponding to tame integrable modules; their levels are
2m+1 1
b(0.6) K = − , where M ∈ Z , m ∈ Z , gcd(M,4m+2) = 1.
≥1 ≥0
2M 2
As shown in [KW15], Section 6, though the corresponding, via the QHR, representations of the
3(2m+1)
N = 3 superconformal algebra with central charge c(K) = − are not modular invariant
M
for m > 0, the Zwegers like modifications of their characters are.
It turns out that for g = spo the corresponding affine Lie superalgebra g has important
2|3
complementary integrable highestweightmodulesL(Λ),namelythose, forwhichallrootvectors
of g, attached to roots α with (α|α) < 0, act locally nilpotently on L(Λ). Suchbtame g-modules
were studied in [KW16], Section 6.4, and [GK15]. The levels of such non-critical modules are
Kb= −m4+2, where m ∈ Z≥1. Consequently, by (0.3), the corresponding admissiblbe modules
have level
m 1
(0.7) K = − − , where m,M ∈ Z .
≥1
4M 2
4
These g-modules are studied in Section 3. Furthermore, in Section 4 we construct, via the
QHR, the corresponding family of representations of the N = 3 Neveu-Schwarz and Ramond
type subperconformal algebras, of central charge c(K) = 3m − 1 (cf. (0.1)) , where M is a
2M 2
positive odd integer, coprime to m, such that their modified (super)characters span a modular
invariant space (for m = 1 modular invariance holds without modification), see Theorem 4.12.
Next, for g = psℓ , the corresponding affine Lie superalgebra g has a family of tame
2|2
integrable modules of negative integer level m (see Section 6), for which we construct in Section
7,viatheQHR,N = 4superconformalalgebramodules,whosemodifiedb(super)charactersform
a modular invariant family. In Section 8 we study the associated family of principal admissible
modules of level
m
(0.8) K = , where M ∈Z ,
≥1
M
and compute their modified (super)characters and those of their twists. They do not span
a modular invariant space. However, we show in Section 9 (see Theorems 9.6-9.8) that the
modified (super)characters of their QHR, which have central charge c(K) = −6 m +1 , do
M
span a modular invariant space, provided that gcd(M,2m) =1 if m ≤ −2.
(cid:0) (cid:1)
Finally, in Section 10 we consider g = D(2,1;a), a family of 17-dimensional exceptional Lie
superalgebras. The tame integrable g-modules exist only for a of the form a = − p , where
p+q
p,q ∈ Z are coprime. Then the level of such a module is of the form
≥1
b
pqn
(0.9) K = − , where n ∈ Z .
≥1
p+q
We construct four families of tame integrable g-modules of level K, compute their modified
supercharacters (there are actually two types of modifications) and show in Corollary 10.22
that they span an SL2(Z)-invariant space. Appblying the QHR to these g-modu;es, we obtain
in Section 11 a family of positive energy irreducible representations of the big N = 4 Neveu-
Schwarz and Ramond type superconformal algebras with central charge cb(K) = 6pqn, where n
p+q
is a positive integer, whose modified (super)characters span a modular invariant subspace.
In Sections 5 and 12 we consider some other cases when the modular invariance holds
without modification.
1 Some important functions and their transformation properties
Fix a positive integer m. A theta function (=Jacobi form) of rank 1 and degree (=index) m is
defined by the following series:
(1.1) Θ (τ,z,t) = e2πimt qmn2e2πimnz, wherej ∈ Z/2mZ,
j,m
n∈XZ+2jm
which converges in the domain (τ,z,t) ∈ C3,Im τ > 0, to a holomorphic function.
These functions have nice elliptic and modular transformation formulas (cf., e.g., Appendix
in [KW14]. The elliptic transformation formulas are as follows for am ∈ Z and k ∈ Z:
(1.2) Θj,m(τ,z+a,t) = eπijaΘj,m(τ,z,t); Θj,m(τ,z + kτ,t) = q−4km2 e−πikzΘj+k,m(τ,z,t).
m
5
The modular transformation formulas are:
1
(1.3) Θj,m −1 , z , t− z2 = −iτ 2 e−πimjj′Θj′,m(τ,z,t),
τ τ 4τ 2m
(cid:18) (cid:19) (cid:18) (cid:19) j′∈Z/2mZ
X
πij2
(1.4) Θj,m(τ +1,z,t) = e 2m Θj,m(τ,z,t).
We often write Θ (τ,z) = Θ (τ,z,0). Especially important are the celebrated four Jacobi
j,m j,m
theta functions of degree two (we put t = 0 here):
ϑ = Θ +Θ , ϑ = −Θ +Θ , ϑ = Θ +Θ , ϑ = iΘ −iΘ .
00 2,2 0,2 01 2,2 0,2 10 1,2 −1,2 11 1,2 −1,2
Formula (1.2) for m = 2 implies that for a,b = 0 or 1 one has:
(1.5) ϑab(τ,z+1) = (−1)aϑab(τ,z); ϑab(τ,z+τ) = (−1)bq−21e−2πizϑab(τ,z).
Formula (1.3) for m = 2 implies that for a,b = 0 or 1 one has:
(1.6) ϑab −1, z = (−i)ab(−iτ)12eπiτz2ϑba(τ,z);
τ τ
(cid:18) (cid:19)
πi
(1.7) ϑ0a(τ +1,z) = ϑ0b(τ,z), where a 6= b; ϑ1a(τ +1,z) = e4 ϑ1a(τ,z).
A mock theta function of rank 1 and degree m ∈ Z is defined by the following series:
>0
(1.8) Φ[m;s](τ,z ,z ) = e2πimj(z1+z2)+2πisz1qj2m+js, wheres ∈ 1Z,
1 1 2 1−e2πiz1qj 2
j∈Z
X
which converges in the domain (τ,z ,z )∈ C3,Im τ > 0, to a meromorphic function.
1 2
As in [KW14]-[KW16], we will extensively use in the paper the functions
(1.9) Φ[m;s](τ,z ,z ,t) = e2πimt Φ[m;s](τ,z ,z )−Φ[m;s](τ,−z ,−z ) ,
1 2 1 1 2 1 2 1
(cid:16) (cid:17)
and
t
(1.10) Ψa[M,b;,εm′,s;ε](τ,z1,z2,t) = qmMabe2πMim(bz1+az2)Φ[m;s](Mτ,z1 +aτ +ε,z2 +bτ +ε, M),
where M is a positive integer, ε,ε′ = 0 or 1, and a,b ∈ ε′+Z.
2
We will often use notation Φ[m;s](τ,z ,z ) = Φ[m;s](τ,z ,z ,0), and similarly for Ψ.
1 2 1 2
The following property of the functions Φ[m;s] will be useful in the sequel.
Lemma 1.1.
z z t z +1 z −1 t
2Φ[m;s](2τ,z ,z ,t) = Φ[2m;2s](τ, 1, 2, )+(−1)2sΦ[2m;2s](τ, 1 , 2 , ).
1 2
2 2 2 2 2 2
Proof. A straightforward verification.
6
ThefunctionsΦ[m;s] andΦ[m;s] haveneithergoodelliptictransformationnormodulartrans-
1
formation properties. In order to achieve these properties one needs to add to these functions
a real analytic correction, defined as follows [Z08], [KW15]. Let E(x) = 2 xe−πu2du. For each
0
j ∈ 1Z let
2 R
R (τ, v) :=
j;m
sgn(n− 1 −j +2m) −E n−2mImv Imτ e−π2imn2τ+2πinv.
2 Imτ m
nX∈12Z (cid:18) (cid:18) (cid:19)r !!
n≡jmod2m
This series converges to a real analytic function for all v,τ ∈ C, Im τ > 0.
Introduce the real analytic “correcting” function
s+2m−1
1 z −z
(1.11) Φ[m;s](τ,z ,z ,t) = e2πimt R τ, 1 2 (Θ −Θ )(τ,z +z ),
add 1 2 2 j;m 2 −j,m j,m 1 2
j=s (cid:18) (cid:19)
X
and let
(1.12) Φ[m;s] := Φ[m;s]+Φ[m;s]
add
be the modification of the function Φ[m;s].
e
The function Φ[m;s] has the following elliptic and modular transformation properties [Z08],
[KW15]:
e
(1.13) Φ[m;s](τ,z +a,z +b,t) = Φ[m;s](τ,z ,z ,t) if a,b ∈ Z,
1 2 1 2
(1.14) Φ[m;s](τ,ze +aτ,z +bτ,t) = q−mabee−2πim(bz1+az2)Φ[m;s](τ,z ,z ,t) if a,b ∈ Z,
1 2 1 2
e 1 z z z z e
(1.15) Φ[m;s] − , 1, 2,t− 1 2 = τ Φ[m;s](τ,z ,z ,t),
1 2
τ τ τ τ
(cid:18) (cid:19)
e e
(1.16) Φ[m;s](τ +1,z ,z ,t) = Φ[m;s](τ,z ,z ,t).
1 2 1 2
[m;s] [m;s] [m;s] [m;s]
Remark 1.2. Properties (1.e13)–(1.16) hold for Φ1 e = Φ1 +Φ1,add, where Φ1,add is given by
(1.11) with Θ −Θ replaced by Θ .
j,m −j,m j,m
[M,m,s;ε] e [M,m,s;ε]
Introduce the modification Ψ of the function Ψ , defined by (1.10), by re-
a,b;ε′ a,b;ε′
placingΦ[m;s] intheRHSof (1.10)byitsmodificationΦ[m;s]. Thenitisnothardtodeducefrom
e
(1.15), (1.16) the following modular transformation properties of these modifications, where ε,
ε′ = 0 or 1, and j,k ∈ ε′ +Z/MZ, provided that Meis odd and coprime to m if m > 1 (see
2
[KW15], Theorem 2.8):
(1.17)
Ψj[M,k,;mε′,s;ε] −τ1,zτ1,zτ2,t− z1τz2 = Mτ e−2πMim(ak+bj) Ψ[aM,b,;mε,s;ε′](τ,z1,z2,t),
(cid:18) (cid:19) a,b∈ε+Z/MZ
X
e e
(1.18) Ψ[jM,k,;mε′,s;ε](τ +1,z1,z2,t) = e2πMimjk Ψ[M,m,s;|ε−ε′|](τ,z1,z2,t).
e e
7
Remark 1.3. Since Θ = Θ for j ∈ Z, we have: Φ[1;s] = Φ[1;s] for s ∈ Z. Consequently,
j,1 −j,1
Φ[1;s] is independent of s ∈ Z. By the denominator identity for sl (see, e.g. [KW94] or
2|1
[KW16], Remark 6.21), we have e
b
η(τ)3ϑ (τ,z +z )
Φ[1;0](τ,z ,z ) = Φ[1;0](τ,z ,z ) = −i 11 1 2 .
1 2 1 2
ϑ (τ,z )ϑ (τ,z )
11 1 11 2
e
[M,1;0;ε] [M,1;0;ε]
Hence from the definition (1.10) we obtain for any positive integer M that Ψ = Ψ
j,k;ε′ j,k;ε′
and
e
Ψj[M,k;,ε1′;0;ε](τ,z1,z2)= −iqMjke2Mπi(kz1+jz2)ϑ η(M(Mτ,τz)3ϑ+11j(τM+τ,εz)ϑ1+(zM2+τ,(zj ++kk)ττ−) ε).
11 1 11 2
e
2 Affine Lie superalgebras and their integrable and admissible modules
In complete analogy with simple Lie algebras, given a basic Lie superalgebra g, we associate
to it the affine Lie superalgebra g = g[t,t−1]+CK+Cd with the same commutation relations.
Choosing a Cartan algebra h of g , we let the Cartan subalgebra of g be h = h+CK+Cd.
¯0
Furthermore, we choose a subsebt of positive roots ∆+ in the set of roots of ∆ of g, such that
the highest root θ is even, and extend the invariant bilinear form (.b|.) onb g, normalized by
condition (0.2), to an invariant bilinear form (.|.) on g in the same way as in the affine Lie
algebra case. Then the restriction of this bilinear form to h is non-degenerate, hence we can
identify h∗ with h. The element of h∗, corresponding to Kb (resp. d) under this identification, is
denoted by δ (resp. Λ ). We use the following coordinates obn h = h∗ :
0
b b b
(2.1) h = 2πi(−τΛ0+z+tδ), where Im τ > 0,bz ∈ hb, t ∈ C.
Let ∆ be the set of roots of g. The corresponding to the choice of the set of positive roots
∆ of g, the set of positive roots of g is ∆ = ∆ ∪( ∪ {α+nδ}), and the corresponding
+ + +
b b αn∈∈∆Z>∪00
subalgebra of g is n+ = n++g[t]t, whberebn+ is the subalgebra of g, corresponding to ∆+. The
set of simple roots is Π = {α = δ−θ}∪Π, where Π = {α ,...,α } is the set of simple roots
0 1 n
for ∆+. As inbthe baffine Lie algebra case, we let ρ= h∨Λ0+ρ, where ρ= 12 α∈∆+(−1)p(α)α.
b
Recall that the Weyl group W of g is the subgroup of GL(h), generated by reflections in the
P
roots α ∈ ∆, such that (α|α) 6= 0. One has W =bW ⋉T , where W is the (finite) Weyl group
L
of g¯0 and TL is the group, conscistingbof translations tβ,β ∈ Lb, where L is the coroot lattice of
g , and, forbβ ∈ h = h∗ the translation t is cdefined by
¯0 β
(2.2) t (Λ) = Λ+(Λ|δ)β − (Λ|β)+ 1(Λ|δ)|β|2 δ, Λ∈ h∗.
β 2
(cid:0) (cid:1)
Given Λ ∈ h∗, one defines, as in the affine Lie algebra case, the irrbeducible highest weight
g-module L(Λ) by the property that it admits a non-zero even weight vector v with weight Λ,
Λ
such that n v b= 0.
+ Λ
b A root α of g is called integrable for L(Λ) if it is even and the root spaces gα and g−α
act locallybnilpotently on L(Λ). A g-module L(Λ) is called integrable (resp. complementary
integrable) if all rboots α of g, such that (α|α) > 0 (resp. (α|α) < 0) are integrable fbor L(Λ)b.
b
b 8
We define the normalized character ch+ and supercharacter ch− of the g-module L(Λ) by
Λ Λ
the series
b
(2.3) ch±(τ,z,t) = qmΛtr± eh,
Λ L(Λ)
wherehisasin(2.1),tr+ andtr− denotethetraceandsupertracerespectively,m = |Λ+ρ|2 −
Λ 2(K+h∨)
sdimg, and K = Λ(K) is the level of Λ. We shall always assume that Λ is non-criticalb, i.e.
24
K +h∨ 6= 0. The series (2.2), given by the weight space decomposition of L(Λ), converge to a
holomorphic function in the domain {h ∈ h| Re α (h) > 0, α ∈ Π}. The simple proof of this
i i
fact, as well as more background in the affine Lie algebra case, may be found in [K90] (the
proof in the affine Lie superalgebra case isbthe same). b
BytheGorelik-Kactheorem,thesefunctionsextendtomeromorphicfunctionsinthedomain
{(τ,z,t)|Im τ > 0} in the affine Lie algebra case, but this is not known for an arbitrary Λ in
the affine Lie superalgebra case (though this holds in all examples at hand).
For tame integrable (andcomplementary integrable, if g = spo ) g-modules L(Λ) one
2m|2m+1
has the conjectural Kac-Wakimoto (super)character formula [KW01], [GK15]:
b
(2.4) R±ch±Λ = jΛ−1q−2(|KΛ++ρbh|∨2) ε±(w)w e(Λ1+±ρb e−β).
β∈T
wX∈W#
b Q
Here W# is the subgroup of the group W, gecnerated by reflections with respect to the roots
α ∈ ∆, which are integrable for L(Λ), j is a positive integer, T is a set, consisting of defect(g)
Λ
positivce isotropic pairwise orthogonal rocots, such that (Λ+ρ|T) = 0, and ε (w) = (−1)ℓ±(w),
±
wherbeℓ (w) (resp. ℓ (w)) isthenumberof reflections r withrespecttoeven (resp. indivisible
+ − α
even) roots α∈ ∆ in a decomposition of w. Finally, R± is thbe normalized (super)denominator:
(2.5) R± = qsdi2m4 geρ (1−(∓1b)p(α)e−α)(−1)p(α).
b
αY∈∆+
b
Formula (2.4) has been checked in [GK15]bin many, but not all cases, considered in this paper.
Denote by ∆# the set of even roots α of g, such that both α and δ−α are integrable (resp.
complementary integrable) for L(Λ), let L# be the lattice spanned over Z by α∨ = 2α/(α|α),
where α ∈ ∆#; this lattice is positive (resp. negative) definite. One has: W# = W# ⋉t ,
L#
where W# is a (finite) subgroup of the group W#, generated by reflections with respect to
exactly one integrable root from the set {α,δ −α}, and t = {t |α ∈ L#c}. Let W# be the
L# α 0
subgroup of W#, generated by reflections r ,α ∈c∆#, and let W# be the corresponding affine
α 0
Weyl group, i.e. the group, generated r and r ,α ∈ ∆#; equivalently, W# = W# ⋉t .
α δ−α 0 0 L#
The integer j in (2.4) can be computed using the following. c
Λ
c
Proposition 2.1. Let
eΛ+ρ
B′ = ε (w)w ,
Λ − (1−b e−β)
wX∈W0# β∈T
Q
and j′ be the coefficient of eΛ+ρ in B′c. Then
Λ Λ
b
j′ = ε (w),
Λ −
w
X
9
#
where the summation is taken over w ∈ W , such that w(Λ+ρ)= Λ+ρ and w(T) > 0.
0
Proof. Each term in the sum, defining B′ is, up to a sign, of the form
cΛ b b
eyt−γ(Λ+ρ)
, where γ ∈ L#,y ∈ W#.
(1−e−y(bβ)q(β|γ)) 0
β∈T
We expand this in a foQrmal power series in qa and e−ω, where a > 0 and ω ∈ ∆ , by the
+
following rules:
1
= e−jωqja if a > 0, or a = 0, ω > 0,
1−e−ωqa
j∈XZ≥0
1 eωq−a
= − = − ejωq−ja if a < 0, or a = 0, ω < 0.
1−e−ωqa 1−eωq−a
j∈XZ≥1
Then we obtain that B′ is a sum of terms of the following form
Λ
±eΛ+ρ+(yt−γ(Λ+ρ)−Λ−ρ)− β∈Tcβy(β)q β∈Tcβ(β|γ),
P P
b b b
wherey ∈ W#,γ ∈ L#,c ∈ C.Butyt (Λ+ρ)−(Λ+ρ)) ∈ L#+Qδ and c y(β) ∈y(T).
0 β −γ β∈T β
Sincethe lattice L# is positive (or negative) definiteand (T|T) = 0, theintersection of L# with
P
y(T) is zero, hence the above term can be eqbual to eΛb+ρ only if w fixes Λ+ρ and all c = 0.
β
This implies the formula for j′ . b
Λ
b
Using (2.2), formula (2.4) can be rewritten as follows:
(2.6) R±ch± = j−1 ε (w)w(Θ± ),
Λ Λ ± Λ+ρ,T
wX∈W#
b
b
where, in coordinates (2.1), we have
(2.7) Θ± (τ,z,t) = e2πi(K+h∨)t q21(K+h∨)|γ|2e2πi(K+h∨)γ(z) .
λ,T (1±q−(γ|β)e−2πiβ(z))
γ∈K+Xλ¯h∨+L# β∈T
Q
Here λ¯ is the restriction of λ ∈ h∗ of level K + h∨ to h. The function (2.7) is a mock theta
function of defect = defect (g) [KW14]. Note that the series (2.7) converges to a meromorphic
function for Im τ > 0, provided thbat (K+h∨)|γ|2 > 0 in (2.7), which will hold in all examples.
(In general, there could be an extra sign in front of each fraction (these are called the signed
mock theta functions), but in all cases, considered in this paper the sign is +.)
Of course, for |T| = 0, (2.7) is a classical Jacobi theta function. For |T| = 1 and |L#| = 1,
(2.7) is an Appell function [Ap].
Now we turn to the description of admissible weights, associated to a given integrable or
complementary integrable weight Λ0, cf. [KW88], [KW15]. First, recall that a subset S of ∆
+
is called simple if
α−β ∈/ ∆ for α,β ∈ S; QS = Q∆. b
b b
10